Pacific Journal of Mathematics AN OPTIMAL SYSTOLIC INEQUALITY FOR CAT(0) METRICS IN GENUS TWO MIKHAIL G. KATZ AND STEPHANE´ SABOURAU Volume 227 No. 1 September 2006 PACIFIC JOURNAL OF MATHEMATICS Vol. 227, No. 1, 2006 AN OPTIMAL SYSTOLIC INEQUALITY FOR CAT(0) METRICS IN GENUS TWO MIKHAIL G. KATZ AND STÉPHANE SABOURAU We prove an optimal systolic inequality for CAT(0) metrics on a genus 2 surface. We use a Voronoi cell technique, introduced by C. Bavard in the hyperbolic context. The equality is saturated by a flat singular metric in the conformal class defined by the smooth completion of the curve y2 = x5 − x. Thus, among all CAT(0) metrics, the one with the best systolic ratio is composed of six flat regular octagons centered at the Weierstrass points of the Bolza surface. 1. Hyperelliptic surfaces of nonpositive curvature Over half a century ago, a student of C. Loewner’s named P. Pu [1952] presented in this journal the first two optimal systolic inequalities, which came to be known as the Loewner inequality for the torus and Pu’s inequality for the real projective plane. (See (5–2) on page 104 for the latter.) The last couple of years have seen the discovery of a number of new systolic inequalities [Ammann 2004; Bangert and Katz 2003; 2004; Bangert et al. 2005; 2006a; 2006b; Ivanov and Katz 2004; Katz 2006; Katz and Lescop 2005; Katz and Sabourau 2006; Katz et al. 2006; Sabourau 2004], as well as near-optimal asymptotic bounds [Hamilton 2005; Katz 2003; Katz and Sabourau 2005; Katz et al. 2005; Rudyak and Sabourau ≥ 2006; Sabourau 2006; ≥ 2006]. A number of questions posed in [Croke and Katz 2003] have thus been answered. A general framework for systolic geometry in a topological context is proposed in [Katz and Rudyak 2005; 2006]. See [Katz ≥ 2006] for an overview of systolic problems. The homotopy 1-systole, denoted sysπ 1(X), of a compact metric space X is the least length of a noncontractible loop of X. Given a metric Ᏻ on a surface, let SR(Ᏻ) denote its systolic ratio sysπ (Ᏻ)2 SR(Ᏻ) = 1 . area(Ᏻ) MSC2000: 53C20, 53C23. Keywords: Bolza surface, CAT(0) space, hyperelliptic surface, Voronoi cell, Weierstrass point, systole. Katz was supported by the Israel Science Foundation (grants no. 620/00-10.0 and 84/03). 95 96 MIKHAIL G. KATZ AND STÉPHANE SABOURAU The optimal systolic ratio of a compact Riemann surface 6 is defined as SR(6) = supᏳ SR(Ᏻ), where the supremum is over all metrics in the conformal type of 6. Finally, given a smooth compact surface M, its optimal systolic ratio is defined by setting SR(M) = sup6 SR(6), where the supremum is over all conformal struc- tures 6 on M. The latter ratio is known for the Klein bottle — see the bound (5–1) on page 104 — in addition to the torus and real projective plane already mentioned. In the class of all metrics without any curvature restrictions, no singular flat metric on a surface of genus 2 can give the optimal systolic ratio in this genus [Sabourau 2004]. The best available upper bound for the systolic ratio of an arbi- trary genus 2 surface is γ2 ' 1.1547 [Katz and Sabourau 2006]. The precise value of SR for the genus 2 surface has so far eluded researchers [Calabi 1996; Bryant 1996]. We propose an answer in the framework of negatively curved, or more generally, CAT(0) metrics. The term “CAT(0) space” evokes an extension of the notion of a manifold of nonpositive curvature to encompass singular spaces. We will use the term to refer to surfaces with metrics with only mild quotient singularities, defined below. Here the condition of nonpositive curvature translates into a lower bound of 2π for the total angle at the singularity. We need such an extension so as to encompass the metric that saturates our optimal inequality (1–1). A mild quotient singularity is defined as follows. Consider a smooth metric 2 2 on ޒ . Let q ≥ 1 be an integer. Consider the q-fold cover Xq of ޒ \{0} with the induced metric. We compactify Xq in the neighborhood of the origin to obtain a c complete metric space Xq = Xq ∪ {0}. c Definition 1.1. Suppose Xq admits an isometric action of ޚp fixing the origin. c Then we can form the orbit space Yp,q = Xq /ޚp. The space Yp,q is then called mildly singular at the origin. The total angle at the singularity is then 2πq/p, and the CAT(0) condition is q/p ≥ 1. Remark 1.2. Alternatively, a point is singular of total angle 2π(1+β) if the metric is of the form eh(z)|z|2β |dz|2 in its neighborhood, where |dz|2 = dx2 + dy2. See [Troyanov 1990, p. 915]. Theorem 1.3. Every CAT(0) metric Ᏻ on a surface 62 of genus 2 satisfies the optimal inequality 1 π 1 √ (1–1) SR(6 , Ᏻ) ≤ cot = 2 + 1 = 0.8047 ... 2 3 8 3 The inequality is saturated by a singular flat metric, with 16 conical singularities, in the conformal class of the Bolza surface. ANOPTIMALINEQUALITYFORCAT(0)METRICSINGENUSTWO 97 The Bolza surface is described in Section 2. The optimal metric is described in more detail in Section 3. Theorem 1.3 is proved in Section 4 based on the octahedral triangulation of S2. Remark 1.4. A similar optimal inequality can be proved for hyperelliptic surfaces of genus 5 based on the icosahedral triangulation [Bavard 1986]. 2. Distinguishing 16 points on the Bolza surface The Bolza surface Ꮾ is the smooth completion of the affine algebraic curve (2–1) y2 = x5 − x. It is the unique Riemann surface of genus 2 with a group of holomorphic automor- phisms of order 48. (A way of passing from an affine hyperelliptic surface to its smooth completion is described in [Miranda 1995, p. 60–61].) Definition 2.1. A conformal involution J of a compact Riemann surface 6 of genus g is called hyperelliptic if J has precisely 2g + 2 fixed points. The fixed points of J are called the Weierstrass points of 6. The quotient Riemann surface 6/J is then necessarily the Riemann sphere, denoted henceforth S2. Let Q : 6 → S2 be the conformal ramified double cover, with 2g + 2 branch points. Thus, J acts on 6 by sheet interchange. Recall that every surface of genus 2 is hyperelliptic, that is, admits a hyperelliptic involution [Farkas and Kra 1992, Proposition III.7.2]. We make note of 16 special points on Ꮾ. We call a point special if it is a fixed point of an order 3 automorphism of Ꮾ. Consider the regular octahedral triangulation of S2 = ރ ∪ ∞. Its set of vertices is conformal to the set of roots of the polynomial x5 −x of formula (2–1), together with the unique point at infinity. Thus the six points in question can be thought of as the ramification points of the ramified conformal double cover Q : Ꮾ → S2, while the 16 special points of Ꮾ project to the eight vertices of the cubic subdivision dual to the octahedral triangulation. In other words, the x-coordinates of the ramification points are {0, ∞, 1, −1, i, −i} , which stereographically correspond to the vertices of a regular inscribed octahe- dron. The conformal type therefore admits the symmetries of the cube. If one includes both the hyperelliptic involution and the real (antiholomorphic) involution of Ꮾ corresponding to the complex conjugation (x, y) → (x¯, y¯) of ރ2, one obtains the full symmetry group Aut(Ꮾ), of order (2–2) |Aut(Ꮾ)| = 96; 98 MIKHAIL G. KATZ AND STÉPHANE SABOURAU see [Kuusalo and Na¨at¨ anen¨ 1995, p. 404] for more details. Lemma 2.2. The hyperbolic metric of Ꮾ admits 12 systolic loops. The 12 loops are in one-to-one correspondence with the edges of the octahedral decomposition of S2. The correspondence is given by taking the inverse image under Q of an edge. The 12 systolic loops cut the surface into 16 hyperbolic triangles. The centers of the triangles are the 16 special points. See [Schmutz 1993, §5] for further details. The Bolza surface is extremal for two distinct problems: • systole of hyperbolic surfaces [Bavard 1992; Schmutz 1993, Theorem 5.2]; • conformal systole of Riemann surfaces [Buser and Sarnak 1994]. The square of the conformal systole of a Riemann surface is also known as its Se- shadri constant [Kong 2003]. The Bolza surface is also conjectured to be extremal for the first eigenvalue of the Laplacian. Such extremality has been verified nu- merically [Jakobson et al. 2005]. The evidence above suggests that the systolically extremal surface may lie in the conformal class of Ꮾ, as well. Meanwhile, we have the following result, proved in Section 5. ≤ π Theorem 2.3. The Bolza surface Ꮾ satisfies SR(Ꮾ) 3 . Note that Theorems 2.3 and 1.3 imply that SR(Ꮾ) ∈ [0.8, 1.05]. 3. A flat singular metric in genus two The optimal systolic ratio of a genus 2 surface (62, Ᏻ) is unknown, but it satisfies the Loewner inequality [Katz and Sabourau 2006]. Here we discuss a lower bound for the optimal systolic ratio in genus 2, briefly described in [Croke and Katz 2003]. The example of M. Berger (see [Gromov 1983, Example 5.6.B0]) in genus 2 is a singular flat metric with conical singularities. It has systolic ratio SR = 0.6666. This ratio was improved by F. Jenni [Jenni 1984], who identified the hyperbolic genus 2 surface with the optimal systolic ratio among all hyperbolic genus 2 sur- faces (see also C.